Critical behavior of dissipative two-dimensional spin lattices

We explore critical properties of two-dimensional lattices of spins interacting via an anisotropic Heisenberg Hamiltonian that are subject to incoherent spin flips. We determine the steady-state solution of the master equation for the density matrix via the corner-space renormalization method. We investigate the finite-size scaling and critical exponent of the magnetic linear susceptibility associated with a dissipative ferromagnetic transition. We show that the von Neumann entropy increases across the critical point, revealing a strongly mixed character of the ferromagnetic phase. Entanglement is witnessed by the quantum Fisher information, which exhibits a critical behavior at the transition point, showing that quantum correlations play a crucial role in the transition.

Figure 1

Angularly averaged magnetic susceptibility ${\chi }_{av}$ versus the normalized coupling parameter $J_y/\gamma$ $(\gamma$ is the dissipation rate) for different sizes of the $L\times L$ lattice. The other coupling parameters are $J_x/\gamma =0.9$ and $J_z/\gamma =1$. Inset: maximum value ${\chi }_{av}^{\max }$ of the susceptibility as a function of the size $L$ of the lattice. The dashed line is a power-law fit of the finite-size scaling. Error bars, when not shown, are smaller than the symbol size.

Figure 2

von Neumann entropy $S$ as a function of the normalized coupling parameter $J_y/\gamma$ for different values of the size $L$ of the squared lattices. Parameters are the same as in Fig. 1. Inset: the derivative of the entropy with respect to the coupling parameter $J_y$.

Figure 3

Quantum Fisher information $F_Q/N$ (normalized by the number $N=L^2$ of sites in the squared lattice) as a function of the normalized coupling parameter $J_y/\gamma$ for different sizes $L$. Parameters are the same as in Fig. 1. The inequality $F_Q/N>1$ witnesses bipartite entanglement. Inset: maximum value of $F_Q/N$ versus the lattice size $L$ (log-log scale) with a power-law fit (dashed line).

Figure 4

Angularly averaged magnetic susceptibility ${\chi }_{av}$ versus normalized coupling parameter $J_y/\gamma$ for different sizes of the $L\times L$ lattice (periodic boundary conditions). Parameters $J_x/\gamma$ and $J_z/\gamma$ are the same as in Fig. 1.

Figure 5

Entanglement negativity $\mathcal{N}$ vs the normalized coupling parameter $J_y/\gamma$ for small lattices. Parameters $J_x/\gamma$ and $J_z/\gamma$ are the same as in Fig. 3.